The analysis below is carried out by dividing the linear spectral statistics \( \sum _{j=1}^{p}f\left( {\bf \lambda }_{j}\right) \)  into two parts: the spiked part \( \sum _{j=1}^{M}f\left( {\bf \lambda }_{j}\right) \)  and the bulk part \( \sum _{j=M+1}^{p}f\left( {\bf \lambda }_{j}\right) \) . From [1]} or [2]} we know the center part of \( \sum _{j=1}^{p}f\left( {\bf \lambda }_{j}\right) \)  is \( p\int f\left( x\right)dF^{c_{n},H_{n}}\left( x\right) \) , where \( c_{n}=\frac{p}{n} \) , \( H_{n}=F^{{\bf \Gamma }} \) , note that here \( {\bf \Gamma }={\bf U}_{2}{\bf D}_{2}{\bf U}_{2}^{\ast } \) . By Cauchy integral formula, \(p\int f\left( x\right)dF^{c_{n},H_{n}}\left( x\right)=-\frac{1}{2\pi i}\oint _{C}f\left(z \right)pm_{1n0}\left(z \right)dz \) , where \( m_{1n0} \)  is the stietjes transform of \( F^{c_{n},H_{n}} \) .
